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Introduction to probability PDF

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Introduction to Probability Second Edition This page is intentionally left blank Introduction to Probability Second Edition George G. Roussas Department of Statistics University of California, Davis AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK OXFORD • PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE SYDNEY • TOKYO Academic Press is an imprint of Elsevier Academic Press is an imprint of Elsevier The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands 225 Wyman Street, Waltham, MA 02451, USA 525 B Street, Suite 1800, San Diego, CA 92101-4495, USA Second edition 2014 Copyright © 2014, 2007 Elsevier Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone ( 44) (0) 1865 + 843830; fax ( 44) (0) 1865 853333; email: [email protected]. Alternatively you can + submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/per- missions, and selecting Obtaining permission to use Elsevier material. Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or prop- erty as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. Library of Congress Cataloging-in-Publication Data Roussas, George G., author. Introduction to probability / by George G. Roussas. -- Second edition. pages cm Includes bibliographical references and index. ISBN 978-0-12-800041-0 1. Probabilities--Textbooks. I. Title. QA273.R8647 2014 519.2--dc23 2013036799 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library For information on all Academic Press publications visit our web site at store.elsevier.com Printed and bound in USA 14 15 16 17 18 10 9 8 7 6 5 4 3 2 1 ISBN: 978-0-12-800041-0 To the memory of David Blackwell and Lucien Le Cam This page is intentionally left blank Contents Preface ��������������������������������������������������������������������������������������������������������xi CHAPTER 1� Some Motivating Examples ����������������������������������������������1 CHAPTER 2� Some Fundamental Concepts �������������������������������������������7 2�1 Some Fundamental Concepts .........................................................7 2�2 Some Fundamental Results ...........................................................11 2�3 Random Variables .........................................................................19 2�4 Basic Concepts and Results in Counting ......................................23 CHAPTER 3� The Concept of Probability and Basic Results ������������31 3�1 Definition of Probability ...............................................................31 3�2 Some Basic Properties and Results ...............................................35 3�3 Distribution of a Random Variable ...............................................44 CHAPTER 4� Conditional Probability and Independence ������������������55 4�1 Conditional Probability and Related Results ................................55 4�2 Independent Events and Related Results ......................................66 CHAPTER 5� Numerical Characteristics of a Random Variable �������79 5�1 Expectation, Variance, and Moment-Generating Function of a Random Variable ...........................................................................79 5�2 Some Probability Inequalities .......................................................89 5�3 Median and Mode of a Random Variable .....................................91 CHAPTER 6� Some Special Distributions ��������������������������������������������99 6�1 Some Special Discrete Distributions ............................................99 6�1�1 Binomial Distribution ........................................................99 6�1�2 Geometric Distribution ....................................................104 6�1�3 Poisson Distribution ........................................................106 6�1�4 Hypergeometric Distribution ...........................................108 6�2 Some Special Continuous Distributions .....................................116 6�2�1 Gamma Distribution ........................................................117 6�2�2 Negative Exponential Distribution ..................................119 6�2�3 Chi-Square Distribution ...................................................121 6�2�4 Normal Distribution .........................................................121 6�2�5 Uniform (or Rectangular) Distribution ............................126 6�2�6 The basics of the Central Limit Theorem(CLT) ..............134 vii viii Contents CHAPTER 7� Joint Probability Density Function of Two Random Variables and Related Quantities ��������������������������������137 7�1 Joint d.f. and Joint p.d.f. of Two Random Variables ...................137 7�2 Marginal and Conditional p.d.f.'s, Conditional Expectation and Variance ................................................................................149 CHAPTER 8� Joint Moment-Generating Function, Covariance, and Correlation Coefficient of Two Random Variables �������������������������������������������������������������������������163 8�1 The Joint m.g.f. of Two Random Variables ................................163 8�2 Covariance and Correlation Coefficient of Two Random Variables .......................................................................167 8�3 Proof of Theorem 1, Some Further Results ................................174 CHAPTER 9� Some Generalizations to k Random Variables, and Three Multivariate Distributions ��������������������������179 9�1 Joint Distribution of k Random Variables and Related Quantities ................................................................179 9�2 Multinomial Distribution ............................................................183 9�3 Bivariate Normal Distribution ....................................................189 9�4 Multivariate Normal Distribution ...............................................198 CHAPTER 10� Independence of Random Variables and Some Applications ������������������������������������������������������������������201 10�1 Independence of Random Variables and Criteria of Independence ..............................................................................201 10�2 The Reproductive Property of Certain Distributions ..................213 10�3 Distribution of the Sample Variance under Normality................222 CHAPTER 11� Transformation of Random Variables������������������������225 11�1 Transforming a Single Random Variable ....................................225 11�2 Transforming Two or More Random Variables ..........................231 11�3 Linear Transformations ...............................................................245 11�4 The Probability Integral Transform ............................................253 11�5 Order Statistics ............................................................................255 CHAPTER 12� Two Modes of Convergence, the Weak Law of Large Numbers, the Central Limit Theorem, and Further Results �����������������������������������������������������265 12�1 Convergence in Distribution and in Probability .........................265 12�2 The Weak Law of Large Numbers and the Central Limit Theorem ............................................................................271 Contents ix 12�2�1 Applications of the WLLN ............................................272 12�2�2 Applications of the CLT ................................................278 12�2�3 The Continuity Correction .............................................280 12�3 Further Limit Theorems ..............................................................288 CHAPTER 13� An Overview of Statistical Inference ������������������������293 13�1 The Basics of Point Estimation ...................................................294 13�2 The Basics of Interval Estimation ...............................................296 13�3 The Basics of Testing Hypotheses ..............................................297 13�4 The Basics of Regression Analysis .............................................301 13�5 The Basics of Analysis of Variance ............................................302 13�6 The Basics of Nonparametric Inference .....................................304 Appendix �������������������������������������������������������������������������������������������������307 Some Notation and Abbreviations �������������������������������������������������������327 Answers to Even-Numbered Exercises �����������������������������������������������331 Index ��������������������������������������������������������������������������������������������������������373

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